Note
Click here to download the full example code
Tikhonov Regularization¶
Tikhonov regularization is a generalized form of L2-regularization. It allows us to articulate our prior knowlege about correlations between different predictors with a multivariate Gaussian prior. Here, we demonstrate how pyglmnet’s Tikhonov regularizer can be used to estimate spatiotemporal receptive fields (RFs) from neural data.
Neurons in many brain areas, including the frontal eye fields (FEF) have RFs, defined as regions in the visual field where visual stimuli are most likely to result in spiking activity.
These spatial RFs need not be static, they can vary in time in a systematic way. We want to characterize how such spatiotemporal RFs (STRFs) remap from one fixation to the next. Remapping is a phenomenon where the RF of a neuron shifts to process visual information from the subsequent fixation, prior to the onset of the saccade. The dynamics of this shift from the “current” to the “future” RF is an active area of research.
With Tikhonov regularization, we can specify a prior covariance matrix to articulate our belief that parameters encoding neighboring points in space and time are correlated.
The unpublished data are courtesy of Daniel Wood and Mark Segraves, Department of Neurobiology, Northwestern University.
# Author: Pavan Ramkumar <pavan.ramkumar@gmail.com>
# License: MIT
Imports
import os.path as op
import numpy as np
import pandas as pd
from pyglmnet import GLMCV
from spykes.ml.strf import STRF
import matplotlib.pyplot as plt
from tempfile import TemporaryDirectory
Download and fetch data files
from pyglmnet.datasets import fetch_tikhonov_data
with TemporaryDirectory(prefix="tmp_glm-tools") as temp_dir:
dpath = fetch_tikhonov_data(dpath=temp_dir)
fixations_df = pd.read_csv(op.join(dpath, 'fixations.csv'))
probes_df = pd.read_csv(op.join(dpath, 'probes.csv'))
probes_df = pd.read_csv(op.join(dpath, 'probes.csv'))
spikes_df = pd.read_csv(op.join(dpath, 'spiketimes.csv'))
spiketimes = np.squeeze(spikes_df.values)
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Design spatial basis functions
n_spatial_basis = 36
n_temporal_basis = 7
strf_model = STRF(patch_size=50, sigma=5,
n_spatial_basis=n_spatial_basis,
n_temporal_basis=n_temporal_basis)
spatial_basis = strf_model.make_gaussian_basis()
strf_model.visualize_gaussian_basis(spatial_basis)
Out:
/Users/mainak/anaconda3/lib/python3.7/site-packages/spykes/ml/strf.py:110: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
plt.show()
Design temporal basis functions
time_points = np.linspace(-100., 100., 10.)
centers = [-75., -50., -25., 0, 25., 50., 75.]
temporal_basis = strf_model.make_raised_cosine_temporal_basis(
time_points=time_points,
centers=centers,
widths=10. * np.ones(7))
plt.plot(time_points, temporal_basis)
plt.show()
Out:
/Users/mainak/Documents/github_repos/pyglmnet/examples/plot_tikhonov.py:78: DeprecationWarning: object of type <class 'float'> cannot be safely interpreted as an integer.
time_points = np.linspace(-100., 100., 10.)
/Users/mainak/Documents/github_repos/pyglmnet/examples/plot_tikhonov.py:85: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
plt.show()
Design parameters
# Spatial extent
n_shape = 50
n_features = n_spatial_basis
# Window of interest
window = [-100, 100]
# Bin size
binsize = 20
# Zero pad bins
n_zero_bins = int(np.floor((window[1] - window[0]) / binsize / 2))
Build design matrix
bin_template = np.arange(window[0], window[1] + binsize, binsize)
n_bins = len(bin_template) - 1
probetimes = probes_df['t_probe'].values
spatial_features = np.zeros((0, n_features))
spike_counts = np.zeros((0,))
fixation_id = np.zeros((0,))
# For each fixation
for fx in fixations_df.index[:1000]:
# Fixation time
fixation_time = fixations_df.loc[fx]['t_fix_f']
this_fixation_spatial_features = np.zeros((n_bins, n_spatial_basis))
this_fixation_spikecounts = np.zeros(n_bins)
unique_fixation_id = fixations_df.loc[fx]['trialNum_f']
unique_fixation_id += 0.01 * fixations_df.loc[fx]['fixNum_f']
this_fixation_id = unique_fixation_id * np.ones(n_bins)
# Look for probes in window of interest relative to fixation
probe_ids = np.searchsorted(probetimes,
[fixation_time + window[0] + 0.1,
fixation_time + window[1] - 0.1])
# For each such probe
for probe_id in range(probe_ids[0], probe_ids[1]):
# Check if probe lies within spatial region of interest
fix_row = fixations_df.loc[fx]['y_curFix_f']
fix_col = fixations_df.loc[fx]['x_curFix_f']
probe_row = probes_df.loc[probe_id]['y_probe']
probe_col = probes_df.loc[probe_id]['x_probe']
if ((probe_row - fix_row) > -n_shape / 2 and
(probe_row - fix_row) < n_shape / 2 and
(probe_col - fix_col) > -n_shape / 2 and
(probe_col - fix_col) < n_shape / 2):
# Get probe timestamp relative to fixation
probe_time = probes_df.loc[probe_id]['t_probe']
probe_bin = np.where(bin_template < (probe_time - fixation_time))
probe_bin = probe_bin[0][-1]
# Define an image based on the relative locations
img = np.zeros(shape=(n_shape, n_shape))
row = int(-np.round(probe_row - fix_row) + n_shape / 2 - 1)
col = int(np.round(probe_col - fix_col) + n_shape / 2 - 1)
img[row, col] = 1
# Compute projection
basis_projection = strf_model.project_to_spatial_basis(
img, spatial_basis)
this_fixation_spatial_features[probe_bin, :] = basis_projection
# Count spikes in window of interest relative to fixation
bins = fixation_time + bin_template
searchsorted_idx = np.searchsorted(spiketimes,
[fixation_time + window[0],
fixation_time + window[1]])
this_fixation_spike_counts = np.histogram(
spiketimes[searchsorted_idx[0]:searchsorted_idx[1]], bins)[0]
# Accumulate
fixation_id = np.concatenate((fixation_id, this_fixation_id), axis=0)
spatial_features = np.concatenate((spatial_features,
this_fixation_spatial_features), axis=0)
spike_counts = np.concatenate((spike_counts,
this_fixation_spike_counts), axis=0)
# Zero pad
spatial_features = np.concatenate((
spatial_features, np.zeros((n_zero_bins, n_spatial_basis))))
fixation_id = np.concatenate((fixation_id, -999. * np.ones(n_zero_bins)))
# Convolve with temporal basis
features = strf_model.convolve_with_temporal_basis(spatial_features,
temporal_basis)
# Remove zeropad
features = features[fixation_id != -999.]
Visualize the distribution of spike counts
plt.hist(spike_counts, 10)
plt.show()
Out:
/Users/mainak/Documents/github_repos/pyglmnet/examples/plot_tikhonov.py:192: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
plt.show()
Plot a few rows of the design matrix
plt.imshow(features[30:150, :], interpolation='none')
plt.show()
Out:
/Users/mainak/Documents/github_repos/pyglmnet/examples/plot_tikhonov.py:198: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
plt.show()
Design prior covariance matrix for Tikhonov regularization
prior_cov = strf_model.design_prior_covariance(
sigma_temporal=3.,
sigma_spatial=5.)
plt.imshow(prior_cov, cmap='Greys', interpolation='none')
plt.colorbar()
plt.show()
np.shape(prior_cov)
Out:
/Users/mainak/Documents/github_repos/pyglmnet/examples/plot_tikhonov.py:208: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
plt.show()
Fit models
from sklearn.model_selection import train_test_split # noqa
Xtrain, Xtest, Ytrain, Ytest = train_test_split(
features, spike_counts,
test_size=0.2,
random_state=42)
from pyglmnet import utils # noqa
n_samples = Xtrain.shape[0]
Tau = utils.tikhonov_from_prior(prior_cov, n_samples)
glm = GLMCV(distr='poisson', alpha=0., Tau=Tau, score_metric='pseudo_R2', cv=3)
glm.fit(Xtrain, Ytrain)
print("train score: %f" % glm.score(Xtrain, Ytrain))
print("test score: %f" % glm.score(Xtest, Ytest))
weights = glm.beta_
Out:
/Users/mainak/Documents/github_repos/pyglmnet/pyglmnet/pyglmnet.py:864: UserWarning: Reached max number of iterations without convergence.
"Reached max number of iterations without convergence.")
train score: 0.048506
test score: 0.015134
Visualize
for time_bin_ in range(n_temporal_basis):
RF = strf_model.make_image_from_spatial_basis(
spatial_basis,
weights[range(time_bin_, n_spatial_basis * n_temporal_basis,
n_temporal_basis)])
plt.subplot(1, n_temporal_basis, time_bin_ + 1)
plt.imshow(RF, cmap='Blues', interpolation='none')
titletext = str(centers[time_bin_])
plt.title(titletext)
plt.axis('off')
plt.show()
Out:
/Users/mainak/Documents/github_repos/pyglmnet/examples/plot_tikhonov.py:246: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
plt.show()
Total running time of the script: ( 6 minutes 12.349 seconds)